Difference between revisions of "1986 AIME Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | What is the sum of the solutions to the equation <math> | + | What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>? |
== Solution == | == Solution == | ||
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* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
| + | {{MAA Notice}} | ||
Revision as of 19:04, 4 July 2013
Problem
What is the sum of the solutions to the equation ![$\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$](http://latex.artofproblemsolving.com/f/4/1/f414e4ec0db945c286625cf56d2b6c7ede93f99d.png)
?
Solution
Let ![$y = \sqrt[4]{x}$](http://latex.artofproblemsolving.com/f/5/8/f5838ae6bb955f1f7afe1c28d870b6cd091b78ac.png)
. Then we have
, or, by simplifying,
![\[y^2 - 7y + 12 = (y - 3)(y - 4) = 0.\]](http://latex.artofproblemsolving.com/f/2/1/f2177dfb0ca3080c826bfa7e508ae0dcd37ef80e.png)
This means that ![$\sqrt[4]{x} = y = 3$](http://latex.artofproblemsolving.com/7/2/3/723fc9924d37616a4675a26e831751451b6904d6.png)
or 
.
Thus the sum of the possible solutions for 
is 
.
See also
| 1986 AIME (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
